Question

In a plane there are $37$ straight lines, of which $13$ pass through the point $A$ and $11$ pass through the point $B$. Besides, no three lines pass through one point, no lines passes through both points $A$ and $B$, and no two are parallel, then the number of intersection points the lines have is equal to

• A. $535$
• B. $601$
• C. $728$
• D. $963$

Answer 1

The correct option is A $535$

Let each line that passes through point A be known as an A line.
Let each line that passes through point B be known as a B line.
Let each line that passes through neither point A nor point B be known as an N line.

Since there 13 A lines, 11 B lines, and a total of 37 lines, the number of N lines $=37-13-11=13$

Case 1: An A line intersects with a B line
Number of options for the A line $=13$
Number of options for the B line $=11$
To combine these options, we multiply:
$13\times 11=143$

Case 2: An N line intersects with an A or B line
Number options for the N line $=13$
Number of options for the A or B line $=13+11=24$
To combine these options, we multiply:
$13\times 24=312$

Case 3: An N line intersects with another N line
Each PAIR of N lines will yield an intersection.
From the 13 N lines, the number of ways to choose$2{=}^{13}{C}_2=(13\times 12)/(2\times 1)=78$

Case 4: Points A and B
Points A and B constitute 2 more intersections $=2$

Total intersections $=143+312+78+2=535$